# Semi-global approach for propagation of the time-dependent Schrödinger equation for time-dependent and nonlinear problems

## Abstract

A detailed exposition of highly efficient and accurate method for the propagation of the time-dependent Schrödinger equation is presented. The method is readily generalized to solve an arbitrary set of ODE's. The propagation is based on a global approach, in which large time-intervals are treated as a whole, replacing the local considerations of the common propagators. The new method is suitable for various classes of problems, including problems with a time-dependent Hamiltonian, nonlinear problems, non-Hermitian problems and problems with an inhomogeneous source term. In this paper, a thorough presentation of the basic principles of the propagator is given. We give also a special emphasis on the details of the numerical implementation of the method. For the first time, we present the application for a non-Hermitian problem by a numerical example of a one-dimensional atom under the influence of an intense laser field. The efficiency of the method is demonstrated by a comparison with the common Runge–Kutta approach.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22701589

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 343; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; LASER RADIATION; NONLINEAR PROBLEMS; ONE-DIMENSIONAL CALCULATIONS; PROPAGATOR; SCHROEDINGER EQUATION; SOURCE TERMS; TIME DEPENDENCE

### Citation Formats

```
Schaefer, Ido, Tal-Ezer, Hillel, and Kosloff, Ronnie.
```*Semi-global approach for propagation of the time-dependent Schrödinger equation for time-dependent and nonlinear problems*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.04.017.

```
Schaefer, Ido, Tal-Ezer, Hillel, & Kosloff, Ronnie.
```*Semi-global approach for propagation of the time-dependent Schrödinger equation for time-dependent and nonlinear problems*. United States. doi:10.1016/J.JCP.2017.04.017.

```
Schaefer, Ido, Tal-Ezer, Hillel, and Kosloff, Ronnie. Tue .
"Semi-global approach for propagation of the time-dependent Schrödinger equation for time-dependent and nonlinear problems". United States. doi:10.1016/J.JCP.2017.04.017.
```

```
@article{osti_22701589,
```

title = {Semi-global approach for propagation of the time-dependent Schrödinger equation for time-dependent and nonlinear problems},

author = {Schaefer, Ido and Tal-Ezer, Hillel and Kosloff, Ronnie},

abstractNote = {A detailed exposition of highly efficient and accurate method for the propagation of the time-dependent Schrödinger equation is presented. The method is readily generalized to solve an arbitrary set of ODE's. The propagation is based on a global approach, in which large time-intervals are treated as a whole, replacing the local considerations of the common propagators. The new method is suitable for various classes of problems, including problems with a time-dependent Hamiltonian, nonlinear problems, non-Hermitian problems and problems with an inhomogeneous source term. In this paper, a thorough presentation of the basic principles of the propagator is given. We give also a special emphasis on the details of the numerical implementation of the method. For the first time, we present the application for a non-Hermitian problem by a numerical example of a one-dimensional atom under the influence of an intense laser field. The efficiency of the method is demonstrated by a comparison with the common Runge–Kutta approach.},

doi = {10.1016/J.JCP.2017.04.017},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 343,

place = {United States},

year = {2017},

month = {8}

}